# A strengthened data processing inequality for the Belavkin-Staszewski relative entropy

## TL;DR

This paper strengthens the data processing inequality for the Belavkin-Staszewski relative entropy, providing new conditions for equality and extending results to a broader class of divergences and quantum channels.

## Contribution

It introduces two new equivalent conditions for equality in the data processing inequality for BS-entropy and extends the results to maximal f-divergences and general quantum channels.

## Key findings

- New conditions characterize equality cases in the data processing inequality.
- Extension of results to maximal f-divergences.
- Application to arbitrary quantum channels via Stinespring dilation.

## Abstract

In this work, we provide a strengthening of the data processing inequality for the relative entropy introduced by Belavkin and Staszewski (BS-entropy). This extends previous results by Carlen and Vershynina for the relative entropy and other standard $f$-divergences. To this end, we provide two new equivalent conditions for the equality case of the data processing inequality for the BS-entropy. Subsequently, we extend our result to a larger class of maximal $f$-divergences. Here, we first focus on quantum channels which are conditional expectations onto subalgebras and use the Stinespring dilation to lift our results to arbitrary quantum channels.